In order to describe general motions and matter in space, functions for angular velocity and density are assumed and denoted Avd, as an abbreviation. The framework provides a unified approach to motions at different scales. It is analysed how Avd enters and rules, in terms of results from equations, in field experiments and observations at Earth. Chaos may organize according to Avd, such that more order, Cosmos, appear in complex nonlinear dynamical systems. This reveals that Avd may be governing and that deterministic systems can be created without assuming boundaries and conditions for initial values and forces from outside. A mathematical model for the initiation of Logos (when a paper accelerates into a narrow circular orbit), was described, and denoted local implosion; Li. The theorem for dl, provides discrete solutions to a power law, and this is related to locations of satellites and moons.

1. Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities
JPAR
Motions for systems and structures in space, described
by a set denoted Avd. Theorems for local implosion; Li,
dl and angular velocities
Lena J-T Strömberg
Previously Department of Solid Mechanics, Royal Institute of Technology, KTH, Sweden
E-mail: lena_str@hotmail.com
In order to describe general motions and matter in space, functions for angular velocity and
density are assumed and denoted Avd, as an abbreviation. The framework provides a unified
approach to motions at different scales. It is analysed how Avd enters and rules, in terms of
results from equations, in field experiments and observations at Earth. Chaos may organize
according to Avd, such that more order, Cosmos, appear in complex nonlinear dynamical
systems. This reveals that Avd may be governing and that deterministic systems can be
created without assuming boundaries and conditions for initial values and forces from
outside. A mathematical model for the initiation of Logos (when a paper accelerates into a
narrow circular orbit), was described, and denoted local implosion; Li. The theorem for dl,
provides discrete solutions to a power law, and this is related to locations of satellites and
moons.
Key words: Angular velocity, density, Avd, Logos, Local implosion, Discrete locations, Theorems, Deterministic Chaos,
Cosmos, unification of kinematics, Lena’s lemma for Riemenn -function
INTRODUCTION
With the purpose of finding origin of motions, and a
proper description without introducing pressure and
forces, a small model will be proposed, and denoted Avd,
as an abbreviation for angular velocity and density. This
is based on functions derived for a non-circular planetary
orbit, Strömberg (2014, 2015). Several applications are
covered in Strömberg (2014, 2015), e.g. planetary
motions, locations, distributed matter, frequencies in
acoustics and occurrence of a factor 3/2, also known
from Correia and Laskar (2004) as the ratio of sidereal
and orbital rotation of Mercury.
If assumed valid at different scales, the model is a
unification of kinematics, which concerns the geometry of
motion in space.
Experiments with natural complex motions, not steady,
monotonic and linear, but where Avd enter, will be
presented and analysed.
Theorems for angular velocities are proposed. To model
the paper experiment, with aggregation occurring with a
large , denoted as a local implosion, a differential
equation is derived. Assuming an iterative solution with
transients, a capture into steady states, corresponding to
e.g. planetary orbits, is obtained. Within the framework,
discrete and quantised solutions are obtained without
assuming a differential equation.
Model for the origin of motions: Avd
Definition. To describe presence of Cosmos, for
dynamical systems in space, a set of functions is defined.
This will be known as Avd.
Avd consists of the functions
i) (t)=0 exp(-2(recc/r0)sin(f0t)) where 0 is constant
Journal of Physics and Astronomy Research
Vol. 2(3), pp. 070-073, September, 2015. © www.premierpublishers.org, ISSN: 2123-503x
Research Article

2. Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities
Strömberg L 070
Figure1. Left. Front view of the paper at small oscillations. Right. After
entering Li, with a rapid rotational velocity
ii) (t)=0 exp (2(recc/r0)sin(f0t)) where 0 is constant
where i) is derived in Strömberg (2015), and ii) is the Le-
density introduced in Strömberg (2014), such that the
continuity of mass, is fulfilled, (i.e. point wise no
expansion or compaction of mass matter.)
Conclusions of physics, for motions, by analysis of
Avd
Conjecture A-Ph (Avd to Physics)
 Elements/Functions in Avd may couple with motion of
a material body in space
 Functional relations for the elements may provide
information on physical laws for motion, and other
properties, e.g. invariants and magnitudes,
 The coupling requires energy, which is supplied by
initial motion of the body in space, and then another
motion or process may begin
The functions in Avd are derived for an orbiting planet
with eccentricity. At Earth, this could be present as a tide,
or a memory of a past non-circular orbit. To be
applicable at a smaller scale, angular velocities are
assumed to be scaled, or that an Avd is created by a
motion in the system, interacting with the surroundings.
Next the results from experiments for a motion will be
presented and analysed.
Avd appearing in a free falling paper (A5, thin)
To obtain knowledge of pure motion, a field experiment of
a natural composed motion not pre-arranged and as little
as possible pertubated, will be analysed in terms of Avd.
It can be described and ruled by an angular velocity,
within short time intervals.
1.First, the paper moves with small horizontal oscillations
c.f. Fig. 1 left, given by the functions 
2.Then it achieves a large angular acceleration. With this
it accelerates into a circular path, where the shape gets
much more curved, Fig. 1 right. The space-time when
curved will be known as initially, Logos, and then Local
implosion, Li.
After one lap, the 'internal energy', in the circular shape is
released, and the paper achieves its original shape.
In the flat shape it moves in a fast translation, in a
different direction. (This may be related to diffusion and
isotropic behaviour in space). At this state, it is also
possible that it obtains degrees of freedom, d.o.f. that we
cannot observe with the eye, (such that presence in other
dimension), coupling with the surrounding, e.g.
Newtonian gravity, and the oscillation at phase 1, or an
own rotational frame created at Logos, however probably
not, since only one lap.
Performing the experiment, you may note that when too
close, the paper approaches towards thee, (in
compaction, contraction, by attraction) and then bumps
outward again.
Sometimes it twists, in oscillations or a twisting lap
instead of a Li, and sometimes it moves in an opposite
horizontal direction before the fast lap.
Detailed description of Logos and Li.
Before Logos, as  increases, the curvature increases,
and at entire revolution, it forms a small circle. This could
be due to either, or all, of the following hypothesis
 It copies the motion at larger scale for Earth
rotation
 An own gravitational field is created
 It obeys a Bernoulli’s law, such that a large
pressure drop balances the increased velocity, and an
isotropic compressive state is prevailed.
A mathematical model for the initiation of Li will be
formulated
Theorem Local implosion: Initiation of the phase 2, for a
falling paper, can be modelled with the functionin Avd,
since these admits high angular acceleration.
Proof. A differentiation of (i), gives the angular
acceleration
d/dt= f00 exp(-2(recc/r0)sin(f0t)) )(-2(recc/r0)cos(f0t))
A linearisation for small f0t close to f0t = gives the
differential equation d/dt=0f02(re/r0), which can be
integrated exactly, to read =0exp(2(re/r0) f0t).
This means that  can increase rapidly, as is the case
when entering Li. Another description which gives an
increased is this format is obtained by assuming that
r0 , i.e. the radius of curvature approaches zero.

3. Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities
J. Phys. Astron. Res. 071
Algebra and Functions on Avd
I. Non-dimensional format
II. Logarithmic ln of the non-dimensional format, gives a
harmonic
III. Power expressions. Solutions to nonlinear differential
equations, and may be the source of forced vibrations.
Result for Earth equator power law
Relations for power expressions, in the formatp
= Cq
will
be known as power laws.
With the conditions at Earth, it will be shown that a power
law is valid for discrete times, or if a spatial description
with t=, discrete angles,.
The relation
2
=G4/3(1)
Is derived from balance of centripetal acceleration and
Newtonian law of gravity, and rules both discrete masses
(i.e. satellites), and arbitrary mass elements parts of
Earth equator considered as a continuum. The result is
experimentally valid without introducing the force, by
measurement of density and angular velocity.
Theorem for discrete (quantised) locations, dl.
Theorem dl. With andfrom Avd, (1) implies discrete
solutions for the locations.
Proof. Identification of the time dependent parts in (1).
The results provide a link between a continuum
description, and the motions of discrete masses e.g.
moons around a planet. If (1) was assumed for Gas
Giants at formation, the number of satellites may be
compared with the close moons for Jupiter (8), and
moons of Saturnus (24).
With angular velocity, the angles may be expressed with
discrete times. Assuming extension in radial direction
provides the orbits and quantisation. To summarize, this
will be denoted the Lena-theorem for dl.
Lena-theorem for Updated angular velocities, Uav.
As an opposite of fast acceleration derived above, we
shall consider an iterative format, which stabilize w into a
constant proportional to. In conjunction with original
format, will also be calculated from previous values, updated
with transients. The ratio , will depend on the
parameters in the functions of Avd, and the Riemann sum
for the Riemann -function.
Preliminaries. Consider a subdivision of the logarithmic
format ln(), into ln(n+1), where n=n-1exp(-
1/g(n)2
), n are integers or rationals and g(n) is a function
of n.
Theorem Uav: Assume from Avd, the harmonic for ln
(n+1)=-2(recc/r0)sin(f0t). The iterative formatn =n-
1exp (-1/n2
), where n are e.g. integers 1,2,3, or half
integers 3/2, 5/2, 7/2,gives a stable constant solution,
such thatn =n-1, for large n. The exact value (for half
integers)is given by
n =0 exp(-2(recc/r0)sin(f0t)-(2)) , whereis the
Riemann-function for half integers.
Lena’s Lemma: relates to the complex Riemann -
function (z)=(1–exp(-zln(2))-1
((z)+1), where z = a+ib
and (z)=(1/nz
)
Proof of Lemma: Evaluation and identification of terms in
the sum.
Proof. Insertion and evaluation of ln of products into sums
of ln, and identification with the Riemann sum.
Remarks.With n being half integers, values more close to
the ratios for gas giants and the planets are obtained,
also for smaller eccentricities.
With the complex Riemann z-function, a harmonic
oscillation is obtained, depending on a dimensionless
parameter b, and with almost constant frequency for
small b. Such couplings may have been used in earlier
calculations and when formulating inventing the Riemann
hypothesis.
Cosmos
Subsequently also the word figure will be used for the
coupling, to manifest that it is something visible which
embodies in a structure.
ConjectureA-(Avd-to-cosmos)
Functions of Avd may appear in other dynamical
systems, e.g. as Cosmos
i) to create order when chaos
ii) to minimize d.o.f.
iii) to obtain a constant energy, or a steady motion
Such cosmos can be included equations and modelling,
as 2
nd
and higher order effects, or as an additional overall
principle or constraint.
Examples. Two clocks at the same wall achieve a
synchronised period. Women living together can achieve
the same evaluation-period, however this could be more
related to the orientation to the heart as a vertical dumb
bell, and the tide.

4. Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities
Strömberg L 072
Figure 2. The motion of the (metallic) flakes is
partly rotational, and downward direction due to
gravity from outside, is not so noticeable.
Additional examples of systems in such motions
Avd figures in dynamical systems, e.g. a free falling
paper. Other examples may be
 Motions of flakes, in a liquid bulb chamber.
When turned upside down, the downward motion due to
gravity, is accomplished with (visually almost dominated
by) much rotation. When bulb is subjected to motion back
and forth around the vertical axis, very much rotation is
transferred to the flakes, such that they rotate rapidly, in
small arcs, c.f. Figure 2.
 Smoke rings in a gravity field, rising since lighter
than air, and form a spiral, consistent with Avd.
 A whirlwind, created over hot sea.
 Logos occurs in flows on curved surfaces, e.g.
the eye globe.
More detailed and extensive models of fluid structure
interaction with whirls are treated in e.g. Walther and
Koumoutsakos (2001) and van Rees et al. (2013).
CONCLUSION
In order to describe general motions and matter in space,
two functions from Strömberg, 2014, 2015), called Avd
were introduced. These were compared with field
experiments for multi-d.o.f. motions, and composed
systems. It was discussed how these may enter and rule
and whether this can be a valid model for many systems
in complex interaction with the surrounding. If so, chaos
may organize according to Avd, and nonlinear
deterministic systems can be described without an ‘a
priori’ assumption of forces and pressure through
boundary conditions. Since applicable at different scales,
Avd provides a unification, not of forces, but of
kinematics.
The same path, but more slow is found for a falling leaf or
feather ‘sailing’ in the air, e.g. in the film Forrest Gump.
A mathematical model for Logos, when the paper starts
accelerates into a narrow circular orbit, denoted Local
implosion, Li, was described. This is characterized by a
large curvature of the paper. If the kinetic energy is
bounded, then the velocity is also bounded, such that a
large angular velocity multiplied with a small radius of the
orbit, is limited and may be initialized with a finite kinetic
energy from previous motion. The word Logos is from
Aristotle (4
th
century BC). For shape memory alloys
Auricchio et al (2008), hard inclusions of martensite are
modelled, with small spheres that may unwarp during
steady state loading. In this context, it can be mentioned
that in classical construction steel, the martensite gives a
harder but more brittle behavior. In some applications
ductile steels may be preferred, since they can withstand
small cracks, and other loads e.g. weld residuals and
environment.
A theorem for dl, provided discrete solutions to a power
law, and this was related to locations of satellites and
moons. From an iterative format, stable constant values
of angular velocities were obtained from a summary of
rationals, derived from results for Riemann -function.

5. Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities
J. Phys. Astron. Res. 073
ACKNOWLEDGEMENTS
To Tech Power Oslo, Stiftelsen Jernkontoret and Dr
Rebbah. The author acknowledged the contributions of
Prof. Marcos Voelzke, Željko Prebeg, Dr. Cyd Ropp and
J. Ponce de Leon for donating their time, critical
evaluation, constructive comments, and invaluable
assistance toward the improvement of this very
manuscript.
REFERENCES
Aristotle 4th century B. Rhetoric, Greek treatise on the art
of persuasion.
Auricchio F, Mielke A, and Stefanelli U (2008).A rate-
independent model for the isothermal quasi-static
evolution of shape-memory materials, Math. Models
Methods Appl. Sci. 18, 125.
Correia A, Laskar J (2004).Mercury's capture into the 3/2
spin-orbit resonance as a result of its chaotic dynamics.
Nature 429, 848-850.
Strömberg L (2014). A model for non-circular orbits
derived from a two-step linearisation of the Kepler laws.
Journal of Physics and Astronomy Research 1(2): 013-
014.
Strömberg L (2015). Models for locations in the solar
system. Journal of Physics and Astronomy Research
1(2): 054-058.
Strömberg L (2014).Generalized potentials describing
orbits in the solar system. Derivation of a ‘close force’
acting on the inner moon Phobos. Journal of
Aerospace Science and Technology 1: 48-52.
van Rees W.M., Gazzola M., Koumoutsakos P
(2013).Optimal shapes for anguilliform swimmers at
intermediate Reynolds numbers. Journal of Fluid
Mechanics, 722.
Walther J.H., Koumoutsakos P (2001).Three-dimensional
vortex methods for particle-laden flows with two-way
coupling, J. Comput. Physics 167: 39-71.
Accepted 15 August, 2015
Citation: Strömberg L (2015). Motions for systems and
structures in space, described by a set denoted
Avd.Theorems for local implosion; Li, dl and angular
velocities. Journal of Physics and Astronomy Research,
2(3): 070-073.
Copyright: © 2015 Strömberg L. This is an open-access
article distributed under the terms of the creative
commons attribution license, which permits unrestricted
use, distribution, and reproduction in any medium,
provided the original author and source are cited.